Scientific Part
Mathematics
Representation of functions on a line by a series of exponential monomials
A. S. Krivosheeva,
O. A. Krivosheevab a Institute of Mathematics with Computing Centre, Ufa Federal Research Center, RAS, 112 Chernyshevsky St., Ufa 450008, Russia
b Bashkir State University, 32 Zaki Validi St., Ufa 450076, Russia
Abstract:
In this work, we consider the weight spaces of integrable functions
$L_p^\omega$ (
$p\geq 1$) and continuous functions
$C^\omega$ on the real line. Let
$\Lambda=\{\lambda_k,n_k\}$ be an unbounded increasing sequence of positive numbers
$\lambda_k$ and their multiplicities
$n_k$,
$\mathcal{E}(\Lambda)=\{t^n e^{\lambda_k t}\}$ be a system of exponential monomials constructed from the sequence
$\Lambda$. We study the subspaces
$W^p (\Lambda,\omega)$ and
$W^0 (\Lambda,\omega)$, which are the closures of the linear span of the system
$\mathcal{E}(\Lambda)$ in the spaces
$L_p^\omega$ and
$C^\omega$, respectively. Under natural constraints on
$\Lambda$ (the finiteness of the condensation index
$S_\Lambda$ and
$n_k/\lambda_k\leq c$,
$k\geq 1$) and on the convex weight
$\omega$, conditions are obtained under which each function of these subspaces continues to an entire function and is represented by a series in the system
$\mathcal{E}(\Lambda)$ that converges absolutely and uniformly on compact sets in the plane. In contrast to the previously known results for the specified representation problem, we do not require that the sequence
$\Lambda$ has a density, and we do not impose the separability condition:
$\lambda_{k+1}-\lambda_k\geq h$,
$k\geq 1$ (instead, the condition of equality to zero of the special condensation index is used).
Key words:
series of exponential monomials, weight space, analytic continuation, condensation index.
UDC:
517.98 Received: 18.03.2022
Revised: 15.04.2022
DOI:
10.18500/1816-9791-2022-22-4-416-429